Can anyone tell me why the impedance is a sloping or irregular line
where all the single band antennas have a very straight trace with the
only blips and dips being connectors, the amp bypass, and the tuner
bypass?

Several thoughts, Roger. First, TDR is usually done with a linear
sweep, typically from 1 MHz to 500 MHz, or whatever the range of the
analyzer.

I'm using the AIM 4170 and just learning how to use the various
functions on it. Being a newcomer to the VNA I'm a complete novice
learning how to use it, let alone how this specific VNA works.

Because the sweep is linear, half of the energy in the sweep
is in the top half of the sweep range, where small discontinuities in a
line are more likely to be detected, but which do NOT relate very well
to the impedance of a connected antenna in the operating range of the
antenna.
Second, the impedance we measure from one end includes reflections from
the other end of the line. If you compare your red and green curves,
their shape is nearly identical -- the same peaks and valleys, with the
major difference between them being that the distance scale is more
compressed. SO -- those peaks and dips are simply a reflection from the
other end of the line.

I kinda though it was related to that, with each band having its own set
of reflections and who knows what other odd reflections show up. Those
reflections vectorially add. Even with a FFT "I think" the result would
still be the equivalent of vectorially adding them.
The last time I used FFTs was for photo enhancement in grad school and
back then it took about 12 hours on the fastest 386 I could find.<:-))

Using a mono-band antenna, without the multi-band antenna, Open,
shorted, or with the 50 ohm standard the line is almost straight with
tiny blips or rather dips showing up for connectors, the six-pack, and
remote antenna switch while the amp and antenna tuner in the bypass
positions show substantial peaks of about 73 ohms each and those do
affect the tuning on 10 and six.

On both 40 and 6 the return loss is high as it should be. On 15 and 10
it's fairly low. I've paid scant attention to the WARC bands so far.

Third, the TDR function in many analyzers (including mine) is an inverse
FFT of the sweep data, and the shape of the trace will depend on the
sweep rate, the excitation signal, how the data is filtered, and how it
is windowed. My VNA, the DG8SAQ VNWA, allows me to change all of those
parameters to maximize the sensitivity to defects.

There are a number of parameters I can change on the AIM 4170C, but as I
said earlier, I'm a complete novice on the steeper part of the learning
curve at present.